Imaginary-Time Evolution

Imaginary-time evolution suppresses high-energy components of a trial state instead of rotating their phases. With nonzero overlap and a spectral gap, normalized evolution approaches the lowest state present in the initial vector. This page develops deterministic propagation; stochastic realizations and their additional biases are treated in diffusion Monte Carlo.

Helpful background. Ground-state projection compares filters, while Rayleigh–Ritz explains representation error.

Exponential filtering and normalization

Use an inverse-energy variable $\tau$, so that

$$|\phi(\tau)\rangle=e^{-\tau(H-E_{\rm ref})}|\phi(0)\rangle.$$

For $|\phi(0)\rangle=\sum_n c_n|n\rangle$, the ratio of an excited coefficient to the ground coefficient is

$$\frac{c_n(\tau)}{c_0(\tau)} =\frac{c_n(0)}{c_0(0)}e^{-\tau(E_n-E_0)}.$$

The scalar shift $E_{\rm ref}$ controls overflow or underflow but cancels from the normalized state. If $c_0(0)=0$, exact propagation never creates ground-state overlap. In a degenerate ground space it selects the normalized projection of the initial state into that space.

Continuous normalization gives a nonlinear but norm-preserving equation,

$$\partial_\tau|\psi\rangle =-(H-\langle H\rangle)|\psi\rangle.$$

For a fixed Hermitian Hamiltonian and normalized state,

$$\frac{d\langle H\rangle}{d\tau} =-2\left(\langle H^2\rangle-\langle H\rangle^2\right)\leq0.$$

Derive this by differentiating the expectation value and inserting the evolution on both sides. The variance is both an energy-descent rate and a stationarity diagnostic. Its vanishing identifies an eigenstate, which need not be the global ground state if symmetry or initial overlap excluded lower levels.

Discrete propagation and its errors

An explicit step $\phi_{k+1}=[I-\delta\tau(H-E_{\rm ref})]\psi_k$, followed by normalization, is a polynomial filter. It is not unconditionally stable. The wanted eigencomponent must have the largest amplification magnitude; excessively large $\delta\tau$ can amplify the top of the spectrum and converge to the wrong state.

Exponential, implicit, split-operator, and Krylov steps can improve stability. If $H=T+V$, a symmetric imaginary-time split uses $e^{-\delta\tau V/2}e^{-\delta\tau T}e^{-\delta\tau V/2}$ before normalization. At fixed nonzero step this propagates an approximation to $H$; extending total projection time cannot remove its time-step bias. Nonlinear normalization does not restore the exact exponential.

Vary total time, step size, and representation independently. Record energy, variance or residual norm, and the target observable. A projection plateau can be caused by a very small gap, ansatz truncation, roundoff, or a fixed-step approximation.

A two-level convergence law

Let $E_0=0$, $E_1=\Delta>0$, and start with equal amplitudes. The normalized excited-state probability is

$$p_1(\tau)=\frac{e^{-2\tau\Delta}}{1+e^{-2\tau\Delta}}, \qquad E(\tau)=\Delta p_1(\tau).$$

To obtain $p_1\leq\varepsilon<1/2$, it is sufficient and necessary here that

$$\tau\geq\frac{1}{2\Delta} \log\frac{1-\varepsilon}{\varepsilon}.$$

This makes the gap dependence concrete: halving the gap doubles the required projection time. The energy variance is $\Delta^2p_1(1-p_1)$ and reproduces the descent equation exactly.

An excited-state calculation can project out previously found states after each step, but their errors and mutual orthogonality must be controlled. See excited-state targeting.

Exercises

For $H=\operatorname{diag}(0,\Delta)$ with $E_{\rm ref}=0$, when does normalized forward Euler suppress the excited component?

Solution

The ground component is multiplied by $1$ and the excited component by $1-\delta\tau\Delta$. Suppression requires $|1-\delta\tau\Delta|<1$, hence $0<\delta\tau\Delta<2$. Between $1$ and $2$ the excited amplitude alternates sign while decaying. At or beyond $2$ it is no longer suppressed.

References

• Bai, Z., J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, eds., Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, SIAM, 2000, online text — filtering and power iteration.
• Leimkuhler, B., and S. Reich, Simulating Hamiltonian Dynamics, Cambridge University Press, 2004 — splitting, modified generators, and numerical integration.